Bounds on the nonnegative signed domination number of graphs

TL;DR

This paper establishes bounds on the nonnegative signed domination number for various classes of graphs, including regular, clique-free graphs, and trees, providing exact characterizations for extremal cases.

Contribution

It introduces new bounds for the nonnegative signed domination number in regular, clique-free graphs, and trees, correcting and extending previous results.

Findings

n/3 is the maximum for cubic graphs of order n

Characterization of graphs where bounds are tight

Bounds for trees and their extremal trees

Abstract

The aim of this work is to investigate the nonnegative signed domination number $\gamma^{NN}_s$ with emphasis on regular, ($r+1$)-clique-free graphs and trees. We give lower and upper bounds on $\gamma^{NN}_s$ for regular graphs and prove that $n/3$ is the best possible upper bound on this parameter for a cubic graph of order $n$, specifically. As an application of the classic theorem of Tur\'{a}n we bound $\gamma^{NN}_s(G)$ from below, for an ($r+1$)-clique-free graph $G$ and characterize all such graphs for which the equality holds, which corrects and generalizes a result for bipartite graphs in [Electron. J. Graph Theory Appl. 4 (2) (2016), 231--237], simultaneously. Also, we bound $\gamma^{NN}_s(T)$ for a tree $T$ from above and below and characterize all trees attaining the bounds.